BME 1450 Term Paper
Tomasz Szwedowski
991555009
1
The Effect of Position Measurement Error on
Finite Element Validation
Tomasz D. Szwedowski

Abstract— Strain measurement is important in the validation
of finite element models of physically loaded structures. The
recent use of strain gages in the study of biomechanical
environments has introduced new concerns that ultimately affect
the quality of the analyses. Specifically, the highly irregular
shapes of biological structures, such as bone, highlight the need
for accurate measurement of strain gage positions. Threedimensional coordinate digitization has emerged as a possibility
for such measurements and warrants a critical analysis of the
errors that may be encountered when using such a technique.
This critique statistically evaluates the ability of such
measurements to accurately reproduce known positions and also
uses an example study to demonstrate the ability of the technique
to produce successful validation of a finite element (FE) model of
the human cranium.
Index Terms— strain gages, experimental error, validation
I. INTRODUCTION
I
n recent decades the methods for the numerical analysis of
stress in physically loaded structures have evolved
enormously. However, these advanced techniques will never
completely replace experimental stress/strain analysis. One of
the most important applications of strain measurement is in the
validation of the numerical methods, among other uses such as
for sensors in active control systems, material property
characterization and the health monitoring of structures. [1]
The purpose of this study was to evaluate the effectiveness
of 3D coordinate digitization as an appropriate method of
transforming the location and vector information of strain
gages into the computational space of the FE model. The goal
being that such an accurate and precise method will allow for
direct comparison of experimental strains with those as
resolved from the FE analyses for the purpose of model
validation.
II. BACKGROUND
Experimental validation of finite element models of
Manuscript received November 7, 2005. T.D. Szwedowski is with the
Orthopaedic Biomechanics Lab at Sunnybrook and Women’s College Health
Sciences centre, 2075 Bayview Avenue, Toronto, On, M4N 3M5. (416)-4805056 email: tommi.szwedowski@utoronto.ca.
physically loaded engineered structures has been
commonplace for many years. FE models introduce
assumptions about the physical phenomenon that is being
considered in the loaded structure and it becomes apparent that
validation of the FE models is paramount in evaluating the
potential of the model for numerical hypothetical studies
especially in biological applications.
The main overriding reason from the development of FE
models can be based on several anecdotal realities. The FE
method allows the resolution of field information throughout
the entire bulk of a structure. A more complete picture is
presented of the physics of the problem than would be seen
from an experimental viewpoint. Strain gage based strain
measurement is limited by the amount of information available
as compared to the FE analysis. Strain gage testing only allows
for a few discrete points, on the surface of the solid structure
of interest.
The deficiencies presented strongly advocate the
development of FE solutions when analyzing complex
structures and loadings. Accurate and precise FE models
become extremely useful in performing hypothesis tests that
would be impractical experimentally. In the case of
biomechanical analyses experimental testing is restricted by
many factors associated with ex vivo testing such as
degradation of the fresh tissue as well as the environment of
the tissue. These in addition to possibilities such as limited
supply of adequate tissue as well as justification for diverting
such tissues from transplantation programs.
Some of the most recent applications of strain gage
experimentation have been in the validation of FE models of
bony structures. Considerable success has been demonstrated
in the development and validation of models of the pelvic,
scapular and metacarpal bones. [2]-[4] In all these studies
strain gages were affixed directly to the cortical surface of the
bone with physiologically relevant loadings applied.
Strain gages operate on the principle that the resistivity of a
wire changes with deformation. It has also been observed that
the relation between the change in resistance and strain is
almost linear. These properties were first observed in 1856 by
Lord Kelvin but were not used for strain measurement until the
1930’s. [5] Modern strain gages are essentially wires etched
out of metallic foil. (See Figure 1)
The 3D coordinate data of the fiducial markers and strain
age vectors were recorded using a MicroScribe™ (Immerson
BME 1450 Term Paper
Corp., San Jose, CA). The 5-axis articulated arm allows the
operator to contact any point in space with a stylus to record
the position. According to company literature, the
MicroScribe™ detects the position based upon optical digital
measurement of the rotations of the articulations. The
company claims that this feature greatly reduces environmental
noise and interference as well as a precision of 0.101 mm.
Figure 1 - Example of typical bonded electrical resistance
strain gage.
III. METHODS
A. Experimental
1) Strain Gage Application
A formalin preserved cadaveric specimen of a complete
human cranium was obtained. The left side of the
craniomaxillofacial complex was dissected away to reveal the
musculature. The masseter and internal pterygoid muscles
were severed at the midpoint and standard 1.0mm sutures were
used in a crosshatch pattern to affix a stainless steel connecting
rod to the severed end of the insertion portion of the masseter.
A 1.0 inch stainless steel rod with flashing was inserted
through the spinal opening and affixed using locking bolts to
the cortical bone. The overlying fat pad and connective tissue
were removed from the surface of the zygomatico-temporal
complex. The exposed cortical bone was de-fatted using
standard strain gage degreaser followed by acetone. The
surface was then abraded using a 320-grit silicon carbide
sanding paper.
Seven 3mm 350  axial or rosette resistance strain gages
(Tokyo Sokki Kenkyujo Co., Tokyo) were bonded to the
specimen using M-100 bonding adhesive (Vishay
Measurement Inc., Malvern, PA). The procedure for
application of the gages was as described in the Vishay
Measurements Group: Manual for Strain Gage technology.[7]
2) Specimen Testing
Tomasz Szwedowski
991555009
2
The prepared specimen was then rigidly fixed to a custom
fabricated fixture with 1 axis of rotation and planar translation
for adjustment of the load direction relative to the zygomaticotemporal complex. The fixture with the specimen in place was
mounted to the static table of an axial/rotational mechanical
testing machine (MTS Corp., Minneapolis, MN).
Load was applied to the specimen through tension
application to the masseter muscle through the sutured steel
rod. The load was provided by the hydraulic cross head of the
MTS through a load cell for load control during the
experiment.
The strain gages were connected to a digital acquisition
device (Iotech, Clevelan, OH). The specimen was loaded by
the MTS with a ramp up to 150N in tension on the masseter
muscle. Strain output was acquired for 7 channels at a
frequency of 25 measurement/s for 20 seconds for a total of
500 data points per load case trial.
The ramp loading was repeated 3 times for each load
direction, for 5 load directions for a total of 15 experimental
trials.
3) 3D Coordinate Digitization
For each loading direction the three-dimensional
coordinates of three fiducial markers, the origin of the strain
gages as well as a second points to define the vector direction
of the strain gage in the experimental space were acquired
using the MicroScribe™.
Using image-processing software (Amira, TGS Berlin)
the coordinates of the fiducial markers were recorded, which
are radiolucent on the CT images of the specimen. This was
possible because CT acquires 3D image data in the form of 2D
cross-sectional slices at regularized intervals in the vertical
direction. The specimen was imaged at a resolution of
512x512 at 1 mm intervals for a total of 415 individual slices.
The rigid affine transformation of the following form was
extracted from the software between the CT and experimental
fiducial markers:
 r11
r
CT
 21
T

Exp
r31

0
r12
r22
r13
r23
r32
0
r33
0
t1 
t 2 
t3 

1
[3]
, where the 3x3 matrix of r-values are the rotation cosines and
the 3x1 column vector is the homogenous translation. This
transformation matrix was then used to map the strain gage
direction vectors from the experimental to the computational
space. The fourth row of the matrix simply accounts for the
combination of rotation and translation and acts as an identity.
The procedure simply involves matrix multiplication with an
additional dimension added to the 3D position vector to satisfy
BME 1450 Term Paper
Tomasz Szwedowski
991555009
the rules of matrix multiplication.
3
Table 1 - Linear regression analysis of the components
between the CT and Transformed fiducial marker
positions.
Table 2- Table 1 - Linear regression analysis of the FE and
experimental strain from the transformed position and
vector data.
B. Finite Element Modeling
A 3D reconstruction of the specimen was developed from
the CT images. The reconstruction was used to generate a FE
mesh to which the experimental boundary and loading
conditions were applied and a novel technique was used to
Table 2 summarizes the results of the regression analysis
map the average pixel value to the individual elements. The between the FE and experimental strains at the locations in the
average pixel values were then scaled to produce the
Y=mx
Regr.
Std.
t
P-value
R
R2
elastic modulus of bone using empirical relations. FE
+b
Error
analyses were performed and the contour tensor data for
.992
.057
17.419
< .0001
.950 .902
m
the surface nodes was extracted.
10.812
2.153
5.021
< .0001
b
The transformed strain gage direction vectors are
directions transformed from the experimental space for all
then used to resolve the FE strains by the following:
loading cases considered at the same time.
 FE  u T Tu
[3]
, where  is the normal strain in the direction of the gage
vector, u is the unit strain gage direction vector and T is
the 3D numerical strain tensor at the origin of the gage.
CT vs. Tranformed Fiducial Coordinates
8
6
C. Statistical Analysis
The experimental fiducial coordinates were transformed
to the computational space using the calculated rigid
transforms for each load case. The statistical analysis of the
data was performed in the SPSS 11.5 software package (SPSS
Inc., Chicago, IL).
The first component of the analysis was to directly compare
the individual coordinate components of the CT fiducial
markers against the same components of the corresponding
experimental fiducial marker coordinates after transformation.
Linear regression analysis was performed in order to assess the
1:1 relationship that should be present between the
components of the coordinate vectors.
After assessment of the quality of the transformation of the
fiducial markers to the computation space the validated
transforms were then used per load case basis to transform the
gage vector directions to the FE space.
These positions and directions were then used to extract the
normal strains as per the method described earlier. These
transformed strains were then compared directly to the
experimental readings obtained during mechanical testing. The
results were fitted with a linear regression model.
4
Rsq = 0.9999
0
Rsq = 0.9985
Load Case 3
-4
Rsq = 1.0000
-6
Load Case 2
Rsq = 1.0000
-8
Load Case 1
-10
-10
Rsq = 0.9999
-8
-6
-4
-2
0
2
6
8
V. DISCUSSION
It should be noted that the linear regression performed in
the analysis of the experimental versus the FE strains is not
an entirely appropriate analysis. In linear regression there is
a clear distinction between the dependent and independent
variables, in that it is assumed that the value of one
continuous variable is dependent on the value of the other.
In FE validation analysis the strains at the gages are defined
as distinct entities not dependent on one another although
linearly related. [6] This means that although linear
regression does give an indication as to the overall trend, the
trend is known before the investigation in that there should
Slope
t-statistic
301.809
463.681
387.089
68.624
327.239
4
Figure 2 – Plot of fiducial marker positions. X-axis is the
CT coordinates while the Y-axis is the transformed
coordinates.
The following are the results obtained from the statistical
analyses of the direct comparison between the individual
components of the CT and transformed experimental position
vectors. Table 1 summarizes the results of the linear regression
analysis between the CT and experimental fiducial marker
position vector components for each loading case.
Load Case 1
Load Case 2
Load Case 3
Load Case 4
Load Case 5
Load Case 4
-2
IV. RESULTS
Linear Regression
Slope
Intercept
.992
-.018
.995
-.013
.994
-.015
.998
-.129
.993
-.017
Load Case 5
2
P-value
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
R
1.000
1.000
1.000
.999
1.000
2
R
1.000
1.000
1.000
.999
1.000
Standard Error
Slope
Intercept
.003
.017
.002
.011
.003
.013
.015
.076
.003
.016
BME 1450 Term Paper
be a one-to-one ratio. As such the correlation coefficient is
much more applicable to the experiment in question.
The results of the analysis on the accuracy of
transformation of the experimental position and vector
direction data are extremely good. The slopes of the
regression curves are between .992 and .998 with r and r 2
values of exactly 1.00 in most cases. (Refer to Table 1) In
addition the significance of these regression fits are
excellent and reflected in the very low P-valued observed
from the hypothesis tests on the slope of the regressions. In
the case of the coordinate transformation accuracy
comparison, linear regression is applicable because the
transformed coordinates are indirectly dependant on the
coordinates of the fiducial markers measured in the CT
images. This is so because the calculation of the rigid
transform requires the CT coordinate data and the fact that
the experimental coordinates are being transformed to the
computational (CT) space.
The second part of the analysis establishes the assessment
of the quality of the measurement of the strain gage position
and vector direction data in its use for the comparison
between the experimentally recorded and the FE strains. It
was deemed insufficient as a route to attaining the goal of
this paper to simply demonstrate that the transformation of
the fiducial markers is accurate but that the resulting data
comparison would also show sufficient agreement for the
sole purpose of assessing the validation of the FE model.
These two notions are distinct because to omit the
assessment of the validation of the FE model would in effect
be a claim that because the fiducial markers are transformed
accurately and precisely that this agreement can be extended
to any relevant data such as the position and vector direction
data.
As such the demonstration of excellent agreement
between the FE and experimental strains is necessary to
establish the 3D coordinate digitization as a gold-standard
method for model validation. The results (Refer to Table 2)
show excellent agreement between the FE and experimental
strains. The correlation coefficients as well as the r2 values
are on par with the results from similar studies of bone. [2][4].
Any discrepancies between the FE and experimental
strains are not attributed to errors in the measurement of the
positions and directions. This is an important outcome, in
that it removes doubt as to the quality of the validation due
to these measurements. This allows investigators to focus on
the issues that do influence the discrepancies ultimately
leading to improvements in model validation practice as a
whole. There are a great number of sources of error that can
be attributed to strain measurement itself but these topics
are beyond the scope of this treatment and will be omitted.
VI. CONLCUSIONS
The statistical analysis has demonstrated that the
Tomasz Szwedowski
991555009
4
transformation of the fiducial markers from the experimental
to the computational space is extremely accurate. Secondly,
the successful agreement between the FE and experimental
strains not only validates the FE models but also the 3D
digitization of strain gage positions and directions.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
S.F. Muller de Almeida, J.S. Hansen. “Enhanced measurement of strain
distributions,” Experimental Mechanics vol. 38, pp. 48-54, Mar. 1998.
A.E. Anderson, C.L. Peters, B.D. Tuttle, J.A. Weiss. “Subject-Specific
Finite Element Model of the Pelvis: Development, Validation and
Sensitivity Studies,” Transactions of the ASME, vol. 127, pp. 364-373,
Jun. 2005.
S. Gupta, F.C.T. van der Helm, J.C. Sterk, F. van Keulen, B.L. Kaptein.
“Development and experimental validation of a three-dimensional finite
element model of the human scapula.” Proc. Instn. Mech. Engrs, Part
H: J. Eng. Med., pp. 127-142.
D.S. Baker, D.J. Netherway, J. Krishnan, T.C. Hearn. “Validation of a
finite element model of the human metacarpal,” Med. Eng. Phys.,
vol.27, pp. 103-113, Jan. 2005.
Strain
Gauges.
(November
2005)
[Online].
Available:
http://www.dur.ac.uk/richard.scott/gauges.html
G.F. Norman, D.L. Striener., Pretty Darned Quick Statistics. London:
BC Decker Inc., 2003, pp. 53-61.
Vishay Measurements Group In.c Strain Gage Technology Manual.
2001.